# Kerodon

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Definition 9.3.1.12. Let $\mathbb {K}$ be a collection of simplicial sets and suppose we are given a commutative diagram of simplicial sets

9.28
$$\begin{gathered}\label{equation:relative-cocompletion-general} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered}$$

where $U$ and $\widehat{U}$ are inner fibrations. We will say that the diagram (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if, for every edge $e: \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the induced map

$H_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$

exhibits $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (in the sense of Definition 9.3.1.8).

If $\kappa$ is a regular cardinal, we say that the diagram (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, where $\mathbb {K}$ is the collection of all $\kappa$-small simplicial sets.